The ring $B(R)$ of all real-valued measurable functions,
carrying the pointwise convergence, is a sequential ring
completion of the subring $C(R)$ of all continuous functions
and, similarly, the ring $\Bbb B$ of all Borel measurable
subsets of $R$ is a sequential ring completion of the
subring $\Bbb B_0$ of all finite unions of half-open
intervals; the two completions are not categorical. We study
$\Cal L_0^*$-rings of maps and develop a completion theory
covering the two examples. In particular, the
$\sigma$-fields of sets form an epireflective subcategory of
the category of fields of sets and, for each field  of sets
$\Bbb A$, the generated $\sigma$-field $\sigma(\Bbb A)$
yields its epireflection. Via zero-rings the theory can be
applied to completions of special commutative $\Cal
L_0^*$-groups.
